PreprintarXiv licencearXiv
An explicit bound on |ζK(1+it)|
Abstract
Let K be a number field of degree n and discriminant Δ. The estimate |ζK(1+it)|≤ CKlogt, where CK is an unspecified constant depending on K, is a well--known result in the literature. We obtain an explicit upper bound for |ζK(1+it)|, depending only on n and Δ. We also show how improvements on the bound depend on improvements on estimates of the ideal counting function IK and the residue of ζK at s=1.
Related
- An explicit lower bound for the unit distance problem
- Optimizing Explicit Unit-Distance Lower-Bound Certificates
- Plotkin-Like Bound and Explicit Function-Correcting Code Constructions for Lee Metric Channels
- An explicit power-saving bound for Sárközy's Theorem on shifted primes, assuming GRH
- An explicit lower bound for the growth exponent of three-dimensional loop-erased random walk
BibTeX
@misc{pelizzari2026explicit,
title = {{An explicit bound on $|ζ_K(1+it)|$}},
author = {Giulia Pelizzari},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32012},
eprint = {2610.32012},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32012},
}
RIS
TY - UNPB
TI - An explicit bound on |ζ_K(1+it)|
AU - Giulia Pelizzari
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32012
UR - https://arxiv.org/abs/2610.32012
AB - Let K be a number field of degree n and discriminant Δ. The estimate |ζ_K(1+it)|≤ C_Klogt, where C_K is an unspecified constant depending on K, is a well--known result in the literature. We obtain an explicit upper bound for |ζ_K(1+it)|, depending only on n and Δ. We also show how improvements on the bound depend on improvements on estimates of the ideal counting function I_K and the residue of ζ_K at s=1.
ER -